Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. HofstadterMy review
rating: 5 of 5 starsWow, this is a difficult book to review.
It is perhaps easier to start by saying that it is not per se about the music of Bach or the art of Escher, and it only discuss the proof of Gödel's incompleteness theorem as an illustration of a much broader and deeper conclusion. To be sure, the book opens with an introduction to the work of Gödel, Escher and Bach before taking an unexpected turn into the world of formal systems for manipulating algebraic expressions. Each chapter is preceded by a fanciful dialogue between Achilles and a Tortoise from Zeno's famous paradox that touches tangentially on a subject or reprises an existing theme, usually aided and abetted by games of structure, word puzzles and some truly atrocious punning. He also touches upon the paradoxes of Zen modes of thinking with self contradictory and (apparently) nonsensical Koans.
Indeed the first half of the book seems to throw in a huge variety of examples, metaphors and ways of thinking before knuckling down to giving a comprehensible proof of Gödel's theorem and then unpacking the consequences of its demolition of the hope of any formal system being necessarily incomplete. It is only in the second half where Hofstadter begins to discuss the basic structures of the human brain and contrasting them with programming languages and systems of artificial intelligence that he tips his hand and reveals that his thesis is that our minds are formal systems. They are highly complex ones but still governed by the same simple rules of logic and thus necessarily incomplete in the Gödelian sense, in that there must be true statements that are ipso facto impossible to prove.
Woah, dude!
At this point you might start to wonder where Hofstadter is going with all of this, but don't panic because he does calm down a little, discussing Turing's famous test for deciding whether a system is intelligent or not and looking forward to the problems that might surround the operation of such systems.
This is a fabulously dense book and it is no surprise that it is taught at degree level (there are some good lecture videos from a course at MIT here). The themes are complex, but one of the benefits of its scattergun use of examples from music, art and mathematics is that most people will find something that they can get a handle on, conceptually speaking, to be able to follow the arguments and ideas under discussion.
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